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Properties

Label 20.20.1984142403...0000.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{20}\cdot 5^{10}\cdot 419^{2}\cdot 691^{4}\cdot 695771^{2}$
Root discriminant $116.11$
Ramified primes $2, 5, 419, 691, 695771$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1030

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![227988105361, 0, -209841439956, 0, 83093153544, 0, -18610574474, 0, 2606127441, 0, -237925889, 0, 14304640, 0, -557393, 0, 13413, 0, -179, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 179*x^18 + 13413*x^16 - 557393*x^14 + 14304640*x^12 - 237925889*x^10 + 2606127441*x^8 - 18610574474*x^6 + 83093153544*x^4 - 209841439956*x^2 + 227988105361)
 
gp: K = bnfinit(x^20 - 179*x^18 + 13413*x^16 - 557393*x^14 + 14304640*x^12 - 237925889*x^10 + 2606127441*x^8 - 18610574474*x^6 + 83093153544*x^4 - 209841439956*x^2 + 227988105361, 1)
 

Normalized defining polynomial

\( x^{20} - 179 x^{18} + 13413 x^{16} - 557393 x^{14} + 14304640 x^{12} - 237925889 x^{10} + 2606127441 x^{8} - 18610574474 x^{6} + 83093153544 x^{4} - 209841439956 x^{2} + 227988105361 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(198414240316397285634688570992640000000000=2^{20}\cdot 5^{10}\cdot 419^{2}\cdot 691^{4}\cdot 695771^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $116.11$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 5, 419, 691, 695771$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{691} a^{14} - \frac{179}{691} a^{12} + \frac{284}{691} a^{10} + \frac{244}{691} a^{8} + \frac{249}{691} a^{6} - \frac{78}{691} a^{4} + \frac{211}{691} a^{2}$, $\frac{1}{691} a^{15} - \frac{179}{691} a^{13} + \frac{284}{691} a^{11} + \frac{244}{691} a^{9} + \frac{249}{691} a^{7} - \frac{78}{691} a^{5} + \frac{211}{691} a^{3}$, $\frac{1}{477481} a^{16} - \frac{179}{477481} a^{14} + \frac{13413}{477481} a^{12} - \frac{79912}{477481} a^{10} - \frac{19790}{477481} a^{8} - \frac{140351}{477481} a^{6} + \frac{36143}{477481} a^{4} + \frac{293}{691} a^{2}$, $\frac{1}{477481} a^{17} - \frac{179}{477481} a^{15} + \frac{13413}{477481} a^{13} - \frac{79912}{477481} a^{11} - \frac{19790}{477481} a^{9} - \frac{140351}{477481} a^{7} + \frac{36143}{477481} a^{5} + \frac{293}{691} a^{3}$, $\frac{1}{3606396087676694090000213975201} a^{18} - \frac{340477939585708754256074}{3606396087676694090000213975201} a^{16} + \frac{323533147426375774964519853}{3606396087676694090000213975201} a^{14} - \frac{265486743533019493783912009091}{3606396087676694090000213975201} a^{12} - \frac{179279659105884690432681302148}{3606396087676694090000213975201} a^{10} - \frac{1631705774539108331308898112260}{3606396087676694090000213975201} a^{8} - \frac{834181769677751071985440252004}{3606396087676694090000213975201} a^{6} + \frac{1250996377725566636253141573}{5219097087809976975398283611} a^{4} + \frac{2557266255836061580818696}{7552962500448591860200121} a^{2} + \frac{3917584645869242290579}{10930481187335154645731}$, $\frac{1}{3606396087676694090000213975201} a^{19} - \frac{340477939585708754256074}{3606396087676694090000213975201} a^{17} + \frac{323533147426375774964519853}{3606396087676694090000213975201} a^{15} - \frac{265486743533019493783912009091}{3606396087676694090000213975201} a^{13} - \frac{179279659105884690432681302148}{3606396087676694090000213975201} a^{11} - \frac{1631705774539108331308898112260}{3606396087676694090000213975201} a^{9} - \frac{834181769677751071985440252004}{3606396087676694090000213975201} a^{7} + \frac{1250996377725566636253141573}{5219097087809976975398283611} a^{5} + \frac{2557266255836061580818696}{7552962500448591860200121} a^{3} + \frac{3917584645869242290579}{10930481187335154645731} a$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $19$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 148111807278000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1030:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 189 conjugacy class representatives for t20n1030 are not computed
Character table for t20n1030 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.911025153125.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 siblings: data not computed
Degree 32 sibling: data not computed
Degree 40 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
419Data not computed
691Data not computed
695771Data not computed